Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 72.6% → 99.2%

Time: 10.7s

Alternatives: 9

Speedup: 1.9×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* (* (* -2.0 J) t_0) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	return ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Fortran

real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    t_0 = cos((k / 2.0d0))
    code = (((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u / ((2.0d0 * j) * t_0)) ** 2.0d0)))
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	return ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	return ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
end

MATLAB

function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)));
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Initial Program: 72.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* (* (* -2.0 J) t_0) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	return ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Fortran

real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    t_0 = cos((k / 2.0d0))
    code = (((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u / ((2.0d0 * j) * t_0)) ** 2.0d0)))
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	return ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	return ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
end

MATLAB

function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)));
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+290}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (* J_s (if (<= t_1 (- INFINITY)) (- U_m) (if (<= t_1 4e+290) t_1 U_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 4e+290) {
		tmp = t_1;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_1 <= 4e+290) {
		tmp = t_1;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / (t_0 * (J_m * 2.0))), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U_m
	elif t_1 <= 4e+290:
		tmp = t_1
	else:
		tmp = U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 4e+290)
		tmp = t_1;
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / (t_0 * (J_m * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U_m;
	elseif (t_1 <= 4e+290)
		tmp = t_1;
	else
		tmp = U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 4e+290], t$95$1, U$95$m]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0

   1. Initial program 4.9%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Simplified 62.6%

      \[\leadsto \color{blue}{J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in J around 0 63.9%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   6. Step-by-step derivation

      1. neg-mul-1 63.9%

         \[\leadsto \color{blue}{-U} \]

   7. Simplified 63.9%

      \[\leadsto \color{blue}{-U} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 4.00000000000000025e290

   1. Initial program 99.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing

   if 4.00000000000000025e290 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 14.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Simplified 62.3%

      \[\leadsto \color{blue}{J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in U around -inf 38.6%

      \[\leadsto \color{blue}{U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 4 \cdot 10^{+290}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := J\_m \cdot \cos \left(K \cdot 0.5\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 1.08 \cdot 10^{-172}:\\ \;\;\;\;\mathsf{fma}\left(-2, t\_0 \cdot \frac{t\_0}{U\_m}, -U\_m\right)\\ \mathbf{else}:\\ \;\;\;\;J\_m \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, U\_m \cdot \frac{0.5}{J\_m}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (* J_m (cos (* K 0.5)))))
   (*
    J_s
    (if (<= J_m 1.08e-172)
      (fma -2.0 (* t_0 (/ t_0 U_m)) (- U_m))
      (* J_m (* (* -2.0 (cos (/ K 2.0))) (hypot 1.0 (* U_m (/ 0.5 J_m)))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = J_m * cos((K * 0.5));
	double tmp;
	if (J_m <= 1.08e-172) {
		tmp = fma(-2.0, (t_0 * (t_0 / U_m)), -U_m);
	} else {
		tmp = J_m * ((-2.0 * cos((K / 2.0))) * hypot(1.0, (U_m * (0.5 / J_m))));
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = Float64(J_m * cos(Float64(K * 0.5)))
	tmp = 0.0
	if (J_m <= 1.08e-172)
		tmp = fma(-2.0, Float64(t_0 * Float64(t_0 / U_m)), Float64(-U_m));
	else
		tmp = Float64(J_m * Float64(Float64(-2.0 * cos(Float64(K / 2.0))) * hypot(1.0, Float64(U_m * Float64(0.5 / J_m)))));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[(J$95$m * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[J$95$m, 1.08e-172], N[(-2.0 * N[(t$95$0 * N[(t$95$0 / U$95$m), $MachinePrecision]), $MachinePrecision] + (-U$95$m)), $MachinePrecision], N[(J$95$m * N[(N[(-2.0 * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U$95$m * N[(0.5 / J$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 1.08e-172

   1. Initial program 64.9%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Simplified 84.2%

      \[\leadsto \color{blue}{J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in J around 0 34.2%

      \[\leadsto \color{blue}{-2 \cdot \frac{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}{U} + -1 \cdot U} \]
   6. Step-by-step derivation

      1. fma-define 34.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}{U}, -1 \cdot U\right)} \]

      2. unpow2 34.2%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\color{blue}{\left(J \cdot J\right)} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}{U}, -1 \cdot U\right) \]

      3. *-commutative 34.2%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\left(J \cdot J\right) \cdot {\cos \color{blue}{\left(K \cdot 0.5\right)}}^{2}}{U}, -1 \cdot U\right) \]

      4. unpow2 34.2%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\left(J \cdot J\right) \cdot \color{blue}{\left(\cos \left(K \cdot 0.5\right) \cdot \cos \left(K \cdot 0.5\right)\right)}}{U}, -1 \cdot U\right) \]

      5. swap-sqr 34.2%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\color{blue}{\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)}}{U}, -1 \cdot U\right) \]

      6. unpow2 34.2%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\color{blue}{{\left(J \cdot \cos \left(K \cdot 0.5\right)\right)}^{2}}}{U}, -1 \cdot U\right) \]

      7. *-commutative 34.2%

         \[\leadsto \mathsf{fma}\left(-2, \frac{{\left(J \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right)}^{2}}{U}, -1 \cdot U\right) \]

      8. neg-mul-1 34.2%

         \[\leadsto \mathsf{fma}\left(-2, \frac{{\left(J \cdot \cos \left(0.5 \cdot K\right)\right)}^{2}}{U}, \color{blue}{-U}\right) \]

   7. Simplified 34.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{{\left(J \cdot \cos \left(0.5 \cdot K\right)\right)}^{2}}{U}, -U\right)} \]
   8. Step-by-step derivation

      1. unpow2 34.2%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)}}{U}, -U\right) \]

      2. associate-/l* 36.1%

         \[\leadsto \mathsf{fma}\left(-2, \color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \frac{J \cdot \cos \left(0.5 \cdot K\right)}{U}}, -U\right) \]

      3. *-commutative 36.1%

         \[\leadsto \mathsf{fma}\left(-2, \left(J \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot \frac{J \cdot \cos \left(0.5 \cdot K\right)}{U}, -U\right) \]

      4. *-commutative 36.1%

         \[\leadsto \mathsf{fma}\left(-2, \left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \frac{J \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}}{U}, -U\right) \]

   9. Applied egg-rr 36.1%

      \[\leadsto \mathsf{fma}\left(-2, \color{blue}{\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \frac{J \cdot \cos \left(K \cdot 0.5\right)}{U}}, -U\right) \]

   if 1.08e-172 < J

   1. Initial program 85.7%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Simplified 95.7%

      \[\leadsto \color{blue}{J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0 77.2%

      \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \color{blue}{0.5 \cdot \frac{U}{J}}\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 77.2%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \color{blue}{\frac{0.5 \cdot U}{J}}\right)\right) \]

      2. *-commutative 77.2%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\color{blue}{U \cdot 0.5}}{J}\right)\right) \]

      3. associate-*r/ 77.2%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \color{blue}{U \cdot \frac{0.5}{J}}\right)\right) \]

   7. Simplified 77.2%

      \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \color{blue}{U \cdot \frac{0.5}{J}}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \left(J\_m \cdot \left(\left(-2 \cdot t\_0\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U\_m}{2}}{J\_m \cdot t\_0}\right)\right)\right) \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* J_s (* J_m (* (* -2.0 t_0) (hypot 1.0 (/ (/ U_m 2.0) (* J_m t_0))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	return J_s * (J_m * ((-2.0 * t_0) * hypot(1.0, ((U_m / 2.0) / (J_m * t_0)))));
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	return J_s * (J_m * ((-2.0 * t_0) * Math.hypot(1.0, ((U_m / 2.0) / (J_m * t_0)))));
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	return J_s * (J_m * ((-2.0 * t_0) * math.hypot(1.0, ((U_m / 2.0) / (J_m * t_0)))))

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	return Float64(J_s * Float64(J_m * Float64(Float64(-2.0 * t_0) * hypot(1.0, Float64(Float64(U_m / 2.0) / Float64(J_m * t_0))))))
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = J_s * (J_m * ((-2.0 * t_0) * hypot(1.0, ((U_m / 2.0) / (J_m * t_0)))));
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * N[(J$95$m * N[(N[(-2.0 * t$95$0), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / 2.0), $MachinePrecision] / N[(J$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 72.5%

   \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

3. Simplified 88.4%

   \[\leadsto \color{blue}{J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 4: 77.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 1.58 \cdot 10^{-170}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;J\_m \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, U\_m \cdot \frac{0.5}{J\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= J_m 1.58e-170)
    (- U_m)
    (* J_m (* (* -2.0 (cos (/ K 2.0))) (hypot 1.0 (* U_m (/ 0.5 J_m))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (J_m <= 1.58e-170) {
		tmp = -U_m;
	} else {
		tmp = J_m * ((-2.0 * cos((K / 2.0))) * hypot(1.0, (U_m * (0.5 / J_m))));
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (J_m <= 1.58e-170) {
		tmp = -U_m;
	} else {
		tmp = J_m * ((-2.0 * Math.cos((K / 2.0))) * Math.hypot(1.0, (U_m * (0.5 / J_m))));
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if J_m <= 1.58e-170:
		tmp = -U_m
	else:
		tmp = J_m * ((-2.0 * math.cos((K / 2.0))) * math.hypot(1.0, (U_m * (0.5 / J_m))))
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (J_m <= 1.58e-170)
		tmp = Float64(-U_m);
	else
		tmp = Float64(J_m * Float64(Float64(-2.0 * cos(Float64(K / 2.0))) * hypot(1.0, Float64(U_m * Float64(0.5 / J_m)))));
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (J_m <= 1.58e-170)
		tmp = -U_m;
	else
		tmp = J_m * ((-2.0 * cos((K / 2.0))) * hypot(1.0, (U_m * (0.5 / J_m))));
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[J$95$m, 1.58e-170], (-U$95$m), N[(J$95$m * N[(N[(-2.0 * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U$95$m * N[(0.5 / J$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if J < 1.58e-170

   1. Initial program 64.9%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Simplified 84.2%

      \[\leadsto \color{blue}{J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in J around 0 35.6%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   6. Step-by-step derivation

      1. neg-mul-1 35.6%

         \[\leadsto \color{blue}{-U} \]

   7. Simplified 35.6%

      \[\leadsto \color{blue}{-U} \]

   if 1.58e-170 < J

   1. Initial program 85.7%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Simplified 95.7%

      \[\leadsto \color{blue}{J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0 77.2%

      \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \color{blue}{0.5 \cdot \frac{U}{J}}\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 77.2%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \color{blue}{\frac{0.5 \cdot U}{J}}\right)\right) \]

      2. *-commutative 77.2%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\color{blue}{U \cdot 0.5}}{J}\right)\right) \]

      3. associate-*r/ 77.2%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \color{blue}{U \cdot \frac{0.5}{J}}\right)\right) \]

   7. Simplified 77.2%

      \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \color{blue}{U \cdot \frac{0.5}{J}}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 69.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 1.55 \cdot 10^{-160}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;J\_m \leq 9.5 \cdot 10^{+109}:\\ \;\;\;\;J\_m \cdot \left(-2 \cdot \mathsf{hypot}\left(1, U\_m \cdot \frac{0.5}{J\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= J_m 1.55e-160)
    (- U_m)
    (if (<= J_m 9.5e+109)
      (* J_m (* -2.0 (hypot 1.0 (* U_m (/ 0.5 J_m)))))
      (* (* -2.0 J_m) (cos (* K 0.5)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (J_m <= 1.55e-160) {
		tmp = -U_m;
	} else if (J_m <= 9.5e+109) {
		tmp = J_m * (-2.0 * hypot(1.0, (U_m * (0.5 / J_m))));
	} else {
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (J_m <= 1.55e-160) {
		tmp = -U_m;
	} else if (J_m <= 9.5e+109) {
		tmp = J_m * (-2.0 * Math.hypot(1.0, (U_m * (0.5 / J_m))));
	} else {
		tmp = (-2.0 * J_m) * Math.cos((K * 0.5));
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if J_m <= 1.55e-160:
		tmp = -U_m
	elif J_m <= 9.5e+109:
		tmp = J_m * (-2.0 * math.hypot(1.0, (U_m * (0.5 / J_m))))
	else:
		tmp = (-2.0 * J_m) * math.cos((K * 0.5))
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (J_m <= 1.55e-160)
		tmp = Float64(-U_m);
	elseif (J_m <= 9.5e+109)
		tmp = Float64(J_m * Float64(-2.0 * hypot(1.0, Float64(U_m * Float64(0.5 / J_m)))));
	else
		tmp = Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5)));
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (J_m <= 1.55e-160)
		tmp = -U_m;
	elseif (J_m <= 9.5e+109)
		tmp = J_m * (-2.0 * hypot(1.0, (U_m * (0.5 / J_m))));
	else
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[J$95$m, 1.55e-160], (-U$95$m), If[LessEqual[J$95$m, 9.5e+109], N[(J$95$m * N[(-2.0 * N[Sqrt[1.0 ^ 2 + N[(U$95$m * N[(0.5 / J$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if J < 1.55e-160

   1. Initial program 64.5%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Simplified 83.7%

      \[\leadsto \color{blue}{J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in J around 0 35.4%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   6. Step-by-step derivation

      1. neg-mul-1 35.4%

         \[\leadsto \color{blue}{-U} \]

   7. Simplified 35.4%

      \[\leadsto \color{blue}{-U} \]

   if 1.55e-160 < J < 9.49999999999999972e109

   1. Initial program 74.4%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Simplified 93.8%

      \[\leadsto \color{blue}{J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0 61.4%

      \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \color{blue}{0.5 \cdot \frac{U}{J}}\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 61.4%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \color{blue}{\frac{0.5 \cdot U}{J}}\right)\right) \]

      2. *-commutative 61.4%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\color{blue}{U \cdot 0.5}}{J}\right)\right) \]

      3. associate-*r/ 61.3%

         \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \color{blue}{U \cdot \frac{0.5}{J}}\right)\right) \]

   7. Simplified 61.3%

      \[\leadsto J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \color{blue}{U \cdot \frac{0.5}{J}}\right)\right) \]

   8. Taylor expanded in K around 0 70.0%

      \[\leadsto J \cdot \left(\color{blue}{-2} \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right) \]

   if 9.49999999999999972e109 < J

   1. Initial program 99.9%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in J around inf 89.9%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 89.9%

         \[\leadsto \color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot -2} \]

      2. *-commutative 89.9%

         \[\leadsto \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot J\right)} \cdot -2 \]

      3. *-commutative 89.9%

         \[\leadsto \left(\cos \color{blue}{\left(K \cdot 0.5\right)} \cdot J\right) \cdot -2 \]

      4. associate-*r* 89.9%

         \[\leadsto \color{blue}{\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot -2\right)} \]

      5. *-commutative 89.9%

         \[\leadsto \cos \color{blue}{\left(0.5 \cdot K\right)} \cdot \left(J \cdot -2\right) \]

      6. *-commutative 89.9%

         \[\leadsto \cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(-2 \cdot J\right)} \]

   7. Simplified 89.9%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(-2 \cdot J\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 1.55 \cdot 10^{-160}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 9.5 \cdot 10^{+109}:\\ \;\;\;\;J \cdot \left(-2 \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 9.5 \cdot 10^{+27}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;J\_m \leq 2.7 \cdot 10^{+45}:\\ \;\;\;\;U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= J_m 9.5e+27)
    (- U_m)
    (if (<= J_m 2.7e+45) U_m (* (* -2.0 J_m) (cos (* K 0.5)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (J_m <= 9.5e+27) {
		tmp = -U_m;
	} else if (J_m <= 2.7e+45) {
		tmp = U_m;
	} else {
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	}
	return J_s * tmp;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (j_m <= 9.5d+27) then
        tmp = -u_m
    else if (j_m <= 2.7d+45) then
        tmp = u_m
    else
        tmp = ((-2.0d0) * j_m) * cos((k * 0.5d0))
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (J_m <= 9.5e+27) {
		tmp = -U_m;
	} else if (J_m <= 2.7e+45) {
		tmp = U_m;
	} else {
		tmp = (-2.0 * J_m) * Math.cos((K * 0.5));
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if J_m <= 9.5e+27:
		tmp = -U_m
	elif J_m <= 2.7e+45:
		tmp = U_m
	else:
		tmp = (-2.0 * J_m) * math.cos((K * 0.5))
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (J_m <= 9.5e+27)
		tmp = Float64(-U_m);
	elseif (J_m <= 2.7e+45)
		tmp = U_m;
	else
		tmp = Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5)));
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (J_m <= 9.5e+27)
		tmp = -U_m;
	elseif (J_m <= 2.7e+45)
		tmp = U_m;
	else
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[J$95$m, 9.5e+27], (-U$95$m), If[LessEqual[J$95$m, 2.7e+45], U$95$m, N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if J < 9.4999999999999997e27

   1. Initial program 66.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Simplified 85.1%

      \[\leadsto \color{blue}{J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in J around 0 36.0%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   6. Step-by-step derivation

      1. neg-mul-1 36.0%

         \[\leadsto \color{blue}{-U} \]

   7. Simplified 36.0%

      \[\leadsto \color{blue}{-U} \]

   if 9.4999999999999997e27 < J < 2.69999999999999984e45

   1. Initial program 53.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in U around -inf 50.1%

      \[\leadsto \color{blue}{U} \]

   if 2.69999999999999984e45 < J

   1. Initial program 99.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in J around inf 84.1%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 84.1%

         \[\leadsto \color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot -2} \]

      2. *-commutative 84.1%

         \[\leadsto \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot J\right)} \cdot -2 \]

      3. *-commutative 84.1%

         \[\leadsto \left(\cos \color{blue}{\left(K \cdot 0.5\right)} \cdot J\right) \cdot -2 \]

      4. associate-*r* 84.1%

         \[\leadsto \color{blue}{\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot -2\right)} \]

      5. *-commutative 84.1%

         \[\leadsto \cos \color{blue}{\left(0.5 \cdot K\right)} \cdot \left(J \cdot -2\right) \]

      6. *-commutative 84.1%

         \[\leadsto \cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(-2 \cdot J\right)} \]

   7. Simplified 84.1%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(-2 \cdot J\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 9.5 \cdot 10^{+27}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.7 \cdot 10^{+45}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.8% accurate, 32.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 2.6 \cdot 10^{+27}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;J\_m \leq 2.55 \cdot 10^{+45}:\\ \;\;\;\;U\_m\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= J_m 2.6e+27) (- U_m) (if (<= J_m 2.55e+45) U_m (* -2.0 J_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (J_m <= 2.6e+27) {
		tmp = -U_m;
	} else if (J_m <= 2.55e+45) {
		tmp = U_m;
	} else {
		tmp = -2.0 * J_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (j_m <= 2.6d+27) then
        tmp = -u_m
    else if (j_m <= 2.55d+45) then
        tmp = u_m
    else
        tmp = (-2.0d0) * j_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (J_m <= 2.6e+27) {
		tmp = -U_m;
	} else if (J_m <= 2.55e+45) {
		tmp = U_m;
	} else {
		tmp = -2.0 * J_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if J_m <= 2.6e+27:
		tmp = -U_m
	elif J_m <= 2.55e+45:
		tmp = U_m
	else:
		tmp = -2.0 * J_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (J_m <= 2.6e+27)
		tmp = Float64(-U_m);
	elseif (J_m <= 2.55e+45)
		tmp = U_m;
	else
		tmp = Float64(-2.0 * J_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (J_m <= 2.6e+27)
		tmp = -U_m;
	elseif (J_m <= 2.55e+45)
		tmp = U_m;
	else
		tmp = -2.0 * J_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[J$95$m, 2.6e+27], (-U$95$m), If[LessEqual[J$95$m, 2.55e+45], U$95$m, N[(-2.0 * J$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if J < 2.60000000000000009e27

   1. Initial program 66.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Simplified 85.1%

      \[\leadsto \color{blue}{J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in J around 0 36.0%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   6. Step-by-step derivation

      1. neg-mul-1 36.0%

         \[\leadsto \color{blue}{-U} \]

   7. Simplified 36.0%

      \[\leadsto \color{blue}{-U} \]

   if 2.60000000000000009e27 < J < 2.5499999999999999e45

   1. Initial program 53.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in U around -inf 50.1%

      \[\leadsto \color{blue}{U} \]

   if 2.5499999999999999e45 < J

   1. Initial program 99.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in J around inf 84.1%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 84.1%

         \[\leadsto \color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot -2} \]

      2. *-commutative 84.1%

         \[\leadsto \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot J\right)} \cdot -2 \]

      3. *-commutative 84.1%

         \[\leadsto \left(\cos \color{blue}{\left(K \cdot 0.5\right)} \cdot J\right) \cdot -2 \]

      4. associate-*r* 84.1%

         \[\leadsto \color{blue}{\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot -2\right)} \]

      5. *-commutative 84.1%

         \[\leadsto \cos \color{blue}{\left(0.5 \cdot K\right)} \cdot \left(J \cdot -2\right) \]

      6. *-commutative 84.1%

         \[\leadsto \cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(-2 \cdot J\right)} \]

   7. Simplified 84.1%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(-2 \cdot J\right)} \]

   8. Taylor expanded in K around 0 46.6%

      \[\leadsto \color{blue}{-2 \cdot J} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 39.9% accurate, 59.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;K \leq 58000:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= K 58000.0) (- U_m) U_m)))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (K <= 58000.0) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (k <= 58000.0d0) then
        tmp = -u_m
    else
        tmp = u_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (K <= 58000.0) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if K <= 58000.0:
		tmp = -U_m
	else:
		tmp = U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (K <= 58000.0)
		tmp = Float64(-U_m);
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (K <= 58000.0)
		tmp = -U_m;
	else
		tmp = U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[K, 58000.0], (-U$95$m), U$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if K < 58000

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Simplified 88.5%

      \[\leadsto \color{blue}{J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in J around 0 30.6%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   6. Step-by-step derivation

      1. neg-mul-1 30.6%

         \[\leadsto \color{blue}{-U} \]

   7. Simplified 30.6%

      \[\leadsto \color{blue}{-U} \]

   if 58000 < K

   1. Initial program 68.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Simplified 87.8%

      \[\leadsto \color{blue}{J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in U around -inf 31.5%

      \[\leadsto \color{blue}{U} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 14.1% accurate, 420.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot U\_m \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m) :precision binary64 (* J_s U_m))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	return J_s * U_m;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    code = j_s * u_m
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	return J_s * U_m;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	return J_s * U_m

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	return Float64(J_s * U_m)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp = code(J_s, J_m, K, U_m)
	tmp = J_s * U_m;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * U$95$m), $MachinePrecision]

Derivation

1. Initial program 72.5%

   \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

3. Simplified 88.4%

   \[\leadsto \color{blue}{J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{2}}{J \cdot \cos \left(\frac{K}{2}\right)}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in U around -inf 27.1%

   \[\leadsto \color{blue}{U} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024118 
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  :precision binary64
  (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))))